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Investigation on principle of polarization-difference imaging in turbid conditions
Optics Communications 413 (2018) 30–38
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Investigation on principle of polarization-difference imaging in turbid conditions Wei Ren, Jinge Guan * School of Information and Communication Engineering, North University of China, Taiyuan 030051, China
a r t i c l e
i n f o
Keywords: Turbid media Polarimetric imaging Optical information processing
a b s t r a c t We investigate the principle of polarization-difference imaging (PDI) of objects in optically scattering environments. The work is performed by both Marius’s law and Mueller–Stokes formalism, and is further demonstrated by simulation. The results show that the object image is obtained based on the difference in polarization direction between the scatter noise and the target signal, and imaging performance is closely related to the choice of polarization analyzer axis. In addition, this study illustrates the potential of Stoke vector for promoting application of PDI system in the real world scene. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Optical imaging through turbid media has important applications in the fields of medical diagnosis, underwater target detection, and driving in the fog. And, it is challenging because light scattering and absorption effects degrade image quality [1–3]. When interactions between the light and the medium are dominated by absorption, imaging performance can be improved by enhancing the illumination energy or the detector sensitivity. However, problems caused by scattering are more serious due to scattered light is superimposed on the target signal to be received by the detector, which is responsible for image blurring and contrast reduction. Image processing and physical model based detection are the two current descattering methods to enhance image quality [4,5]. Compared with the latter, the former is not sufficient to remove scattering effect from images [6]. When using physical model based method to detect and recognize objects in turbid conditions, the principle is based on the difference in physical properties, such as timeof-flight [7,8], polarization [9,10], and coherence [11], between the scatter noise and the target signal. Among the above imaging methods, polarimetric imaging through turbid media is promising due to its advantage of low cost and easy operation. Also, it has attracted more and more attention of researchers because certain problems still need to be solved for the purpose of application in the real world. The research progresses in active polarimetric imaging aspect are mainly as follows. Under polarization illumination, polarization imaging with a cross polarization detection could increase the visibility range in scattering media compared with
direct imaging [12]. In the cross polarization detection method, circular polarization shows a better performance than linear polarization due to the former is depolarized by the object more than the latter [13]. Investigations on the interactions of object and polarized light by Mueller matrix further demonstrate that circular polarization is more depolarized than linear polarization [14]. Asymmetric segmented phaseonly filter is introduced to cross polarization detection method to enhance visibility [15]. Co-polarization image is further acquired, and is combined with cross polarization image to recover the depolarized object in a scattering medium [10]. Optical correlation-based approach is used to improve performance of polarization recovery [16]. And polarization recovery method by estimating the polarized-difference signal can obtain not only depolarized but also polarized objects [17]. For active detection method, the polarization state of light has been known, and is typically under the experimenters’ control [18]. It should be noted that polarization detection with passive mode is seldom concerned. Polarization-difference imaging (PDI) is a bio-inspired technique that uncovers the features of object surface in a scattering environment with ambient light [18]. Quantitative research shows that when compared with direct imaging, PDI increases distance at which certain target features can be detected by 2–3 times [19]. And a combination of direct imaging and PDI is suitable for detecting weakly polarized objects in scattering media, which could be well characterized by colorimetric representations [20]. PDI is further demonstrated to enhance the pointspread function for imaging [21]. Adaptive algorithms are proposed to
* Corresponding author.
E-mail address: [email protected] (J. Guan). https://doi.org/10.1016/j.optcom.2017.12.025 Received 16 September 2017; Received in revised form 7 November 2017; Accepted 10 December 2017 0030-4018/© 2017 Elsevier B.V. All rights reserved.
W. Ren, J. Guan
Optics Communications 413 (2018) 30–38
Fig. 1. Imaging model in the scattering medium: (1) direction transmission, (2) veiling light, (3) light lost due to scattering, (4) light absorbed by the medium.
Fig. 2. Polarization filtering in PDI. 𝑉 : veiling light, 𝐷: direct transmission. 𝑃 𝑆𝐴1 and 𝑃 𝑆𝐴2 are the orthogonal polarizations, respectively.
deal with diversity of polarization direction distributions [22]. Micropolarizer array is used to enhance image contrast for PDI technique by controlling a micropatterned liquid-crystal cell alignment [23]. However, for traditional PDI systems, common-mode rejection of scattering effect is performed by mechanical rotation of the axis of polarization state analyzer, which is helpless in implementing real-time detection. In fact, polarimetric imaging in a scattering medium is based on the difference in polarization responses between the target signal and the scatter noise. In order to improve the performance of PDI technique, the principle of PDI should be understood. This is exactly what we want to study in the paper.
the difference between 45◦ and 135◦ polarizations, 𝑠3 represents the difference between right and left circular polarizations, 𝑇 is the symbol of transpose matrix. Two polarization parameters, i.e. degree of polarization (𝐷𝑜𝑃 ) and angle of polarization (𝜙), can be obtained from the Stokes vector √ 𝑠21 + 𝑠22 + 𝑠23 𝐷𝑜𝑃 = (2) 𝑠0 and 𝜙=
2. Imaging model
(3)
respectively. Degree of polarization indicates the proportion of polarized light in the total light intensity, and angle of polarization represents the polarization direction of polarized light. Light can be classified into completely polarized light (𝐷𝑜𝑃 = 1), un-polarized light (𝐷𝑜𝑃 = 0), and partially polarized light (0 < 𝐷𝑜𝑃 < 1). Laan et al. ever considered the partially polarized light as the sum of completely polarized light and unpolarized light [27]. Here, we further decompose partial polarized light into linearly polarized, circularly polarized, and un-polarized light, as shown in Eq. (4)
For passive optical imaging in a scattering medium with ambient illumination, light undergoing scattering events from the target can be classified into four categories [19,24], as shown in Fig. 1. The direction transmission is the component corresponding to the target. Veiling light comes from particles in front of the target, which is also referred as path radiance [25]. Light being scattered away from the detector and that being absorbed by the medium could not be recorded. Thus we only consider the components of direction transmission and veiling light. The latter contains no target information and causes the degradation of image quality. In this paper, we investigate the principle of PDI based on the difference in polarization properties between the direction transmission and the veiling light.
𝑺 = 𝑺linearly−polarized + 𝑺circularly−polarized + 𝑺un−polarized √ √ ⎡ 𝑠2 + 𝑠2 ⎤ ⎡ ⎤ ⎡𝑠 − 𝑠2 + 𝑠2 − 𝑠 ⎤ 𝑠 3⎥ 0 3 1 2 1 2 ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ 0 𝑠1 0 ⎥+⎢ ⎥+⎢ ⎥. = ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ 0 𝑠 0 2 ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ 0 0 ⎦ ⎣ 𝑠3 ⎦ ⎣ ⎣ ⎦
3. Principle
(4)
On the basis of polarization theory, for both circularly polarized and un-polarized light, their decompositions in arbitrary orthogonal polarization directions equal to each other. Referring to Eq. (1), circular polarization and un-polarized components of the detected light become zero in the process of PDI. Therefore, what the PDI technique actually deals with is the linear polarization component. Fig. 2 gives the physical process of polarization filtering for PDI. We suppose that the angle between the orientation of one polarization axis of the polarization state analyzer (𝑃 𝑆𝐴1 ) and the polarization direction of scatter noise is 𝜃, and the angle between polarization directions of target signal and scatter noise is 𝛿, respectively. According to Eq. (4), √
3.1. Concept of PDI Motivated by the visual system of green sunfish which is sensitive to polarization for serving in imaging [26], two orthogonal polarization images are captured, and the concept of bio-inspired PDI is defined by calculating the difference between these two images [18] 𝐼PD (𝑥, 𝑦) = 𝐼∥ (𝑥, 𝑦) − 𝐼⊥ (𝑥, 𝑦)
𝑠 1 arctan( 2 ) 2 𝑠1
(1)
where symbols of ∥ and ⊥ represent two arbitrary orthogonal polarizations, and (𝑥, 𝑦) indicates the pixel position in the image. In the optical polarization information processing (Eq. (1)), imaging performance is dependent on the choice of polarization axis, and the scatter noise is removed by means of common-mode rejection. Under what circumstances can this descattering method work? With such an object, we further investigate the physical process of this polarization filtering based on the Marius’s law and the Mueller–Stokes formalism.
the linear polarization intensity of light is 𝑠21 + 𝑠22 . And polarization components in the orthogonal polarization directions can be obtained based on Marius’s law. For the scatter noise, its polarization components in the directions of 𝑃 𝑆𝐴1 and 𝑃 𝑆𝐴2 are √ 𝐼∥ (𝑉 ) = 𝑠21 (𝑉 ) + 𝑠22 (𝑉 )(𝑡2𝑥 ⋅ cos 𝜃 + 𝑡2𝑦 ⋅ sin 𝜃) (5)
3.2. Marius’s law based analysis
and Stokes vector, which consists of four elements, is commonly used to describe the polarization state of light. This vector can be expressed by the formula S = [𝑠0 , 𝑠1 , 𝑠2 , 𝑠3 ]T , where 𝑠0 is the total light intensity, 𝑠1 indicates the difference between 0◦ and 90◦ polarizations, 𝑠2 is
𝐼⊥ (𝑉 ) =
√
𝑠21 (𝑉 ) + 𝑠22 (𝑉 )(𝑡2𝑥 ⋅ sin 𝜃 + 𝑡2𝑦 ⋅ cos 𝜃)
(6)
respectively, where 𝑡𝑥 and 𝑡𝑦 are the maximum and minimum transmission coefficients of polarization state analyzer, respectively. 31
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Optics Communications 413 (2018) 30–38
For the target signal, its polarization components in the directions of 𝑃 𝑆𝐴1 and 𝑃 𝑆𝐴2 are √ [ ] (7) 𝐼∥ (𝐷) = 𝑠21 (𝐷) + 𝑠22 (𝐷) 𝑡2𝑥 ⋅ cos 2(𝜃 + 𝛿) + 𝑡2𝑦 ⋅ sin 2(𝜃 + 𝛿) and √ 𝐼⊥ (𝐷) =
𝑠21 (𝐷) + 𝑠22 (𝐷)[𝑡2𝑥 ⋅ sin 2(𝜃 + 𝛿) + 𝑡2𝑦 ⋅ cos 2(𝜃 + 𝛿)]
(8)
respectively. On the basis of Eq. (1), the PDI components of scatter noise and target signal are 𝐼PD (𝑉 ) = 𝐼∥ (𝑉 ) − 𝐼⊥ (𝑉 ) √ = (𝑡2𝑥 − 𝑡2𝑦 ) ⋅ 𝑠21 (𝑉 ) + 𝑠22 (𝑉 ) cos 2𝜃
Fig. 3. Orientation distributions of polarization axis. (a) 0 ≤ 𝛼 ≤ 𝜋∕2. (b) −𝜋∕2 ≤ 𝛼 > 0.
(9)
On the basis of Eqs. (15) and (16), the polarization states of detected light on the two orthogonal polarization directions are expressed as equations in Box II. Since the first element of Stokes vector represents the total intensity of light, the PDI components of scatter noise and target signal are calculated referring to Eqs. (17) and (18), which are shown as
and 𝐼PD (𝐷) = 𝐼∥ (𝐷) − 𝐼⊥ (𝐷) √ = (𝑡2𝑥 − 𝑡2𝑦 ) ⋅ 𝑠21 (𝐷) + 𝑠22 (𝐷) cos 2(𝜃 + 𝛿)
(10)
respectively.
𝐼PD (𝑉 ) = 𝐼∥ (𝑉 ) − 𝐼⊥ (𝑉 ) = (𝑡2𝑥 − 𝑡2𝑦 )[cos 2𝛼 ⋅ 𝑠1 (𝑉 ) + sin 2𝛼 ⋅ 𝑠2 (𝑉 )]
3.3. Mueller–Stokes formalism based analysis
and 𝐼PD (𝐷) = 𝐼∥ (𝐷) − 𝐼⊥ (𝐷)
When light interacts with the object, the relationship between the polarization state of transmitted light and that of incident light can be expressed as
= (𝑡2𝑥 − 𝑡2𝑦 )[cos 2𝛼 ⋅ 𝑠1 (𝐷) + sin 2𝛼 ⋅ 𝑠2 (𝐷)]
where 𝑺out and 𝑺in are Stokes vectors of transmitted light and incident light, respectively, 𝑴 represent the Mueller matrix of the object, which is a 4 × 4 array. On the basis of optical polarization theory, Mueller matrix of the polarization state analyzer with polarization axis orientated at the angle of 𝛼 with respect to the horizontal direction is calculated as 𝑀PSA (𝛼) = 𝐴(−𝛼)𝑀PSA (0◦ )𝐴(𝛼)
3.4. PDI quality evaluation Since the performance of PDI depends on the polarization axis of polarization state analyzer [18], we use the ratio of PDI components of scatter noise and target signal in Sections 3.2 and 3.3 to investigate how the choice of polarization axis affects the image quality, which is expressed as
(12)
where 𝐴(𝛼) corresponds to the 4×4 rotation matrix with 𝛼 varying from −𝜋∕2 to 𝜋/2, and 𝑀𝑃 𝑆𝐴 (0◦ ) is the Mueller matrix of polarization state analyzer with polarization axis orientated at the angle of 0◦ . The formulas of them can be expressed as ⎡1 ⎢ 0 𝐴(𝛼) = ⎢ ⎢0 ⎢0 ⎣
0 cos 2𝛼 − sin 2𝛼 0
0 sin 2𝛼 cos 2𝛼 0
0⎤ ⎥ 0⎥ 0⎥ 1⎥⎦
⎡𝑡2𝑥 + 𝑡2𝑦 ⎢2 𝑡 − 𝑡2𝑦 𝑀PSA (0◦ ) = ⎢ 𝑥 ⎢ 0 ⎢ ⎣ 0
𝑡2𝑥 − 𝑡2𝑦 𝑡2𝑥 + 𝑡2𝑦 0 0
0 0 2𝑡𝑥 𝑡𝑦 0
𝐼PD (𝑉 ) 𝐼PD (𝐷) √ 𝑠21 (𝑉 ) + 𝑠22 (𝑉 ) = √ ⋅ 𝑟(𝜃, 𝛿) 𝑠21 (𝐷) + 𝑠22 (𝐷)
𝑅(𝜃, 𝛿) =
(21)
where the formula of 𝑟(𝜃, 𝛿) is
(13)
𝑟(𝜃, 𝛿) =
and 0 ⎤ ⎥ 0 ⎥ 0 ⎥ ⎥ 2𝑡𝑥 𝑡𝑦 ⎦
(20)
respectively. We have analyzed the physical process of PDI by Marius’s law and Mueller–Stokes formalism, and the contents described by them are identical in essence, which is provided in Appendix
(11)
𝑺out = 𝑀 ⋅ 𝑺in
(19)
cos 2𝜃 . cos 2(𝜃 + 𝛿)
(22)
It can be observed from Eq. (21) that, image quality obtained by the PDI technique is proportional to the value of 𝑅(𝜃, 𝛿). Fig. 4 further gives numerical plots corresponding to the function 𝑟(𝜃, 𝛿), in which both 𝜃 and 𝛿 range from 0 to 𝜋/2. For the purpose of simplified analysis, linear polarization components of 𝑠1 and 𝑠2 are neglected due to they are fixed in the image and not affected by the choice of polarization axis. In Fig. 4(a), the value of 𝑟(𝜃, 𝛿) is affected by the value of 𝜃 when 𝛿 does not equal to zero, and it is always zero with 𝜃 equaling to 𝜋/4. An important feature is that 𝑟(𝜃, 𝛿) always equal to one when 𝛿 equals zero, which illustrates that image quality obtained by PDI technique is nothing to do with the choice of polarization axis. In order to understand the properties of function 𝑟(𝜃, 𝛿) intuitively, two dimensional numerical plots at the case of 𝛿 equaling to 0, 𝜋∕4 and 𝜋∕3, subsequently, are further given in Fig. 4(b)–(d). These results demonstrate that when angles between the scatter noise and each of the orthogonal polarization axis are both 𝜋/4, the scattering effect can be eliminated by means of common-mode rejection. It should also be noted that, common-mode rejection based descattering method is valid only when the value of
(14)
respectively. When using the PDI technique to detect objects in a scattering environment, we suppose that the angle between the orientation of one polarization axis of the polarization state analyzer (𝑃 𝑆𝐴1 ) and the horizontal direction is arbitrary with a value of 𝛼. It can be observed from Fig. 3 that when computing the difference between these two orthogonal polarizations (𝑃 𝑆𝐴1 and 𝑃 𝑆𝐴2 ), results corresponding to Fig. 3(a) and (b) have opposite values. Thus only the case in Fig. 3(a) is analyzed in this paper. Combining Eqs. (12)–(14), Mueller matrices of the polarization state analyzer corresponding to 𝑃 𝑆𝐴1 and 𝑃 𝑆𝐴2 are calculated as equations given in Box I. 32
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⎡ (𝑡2𝑥 + 𝑡2𝑦 ) ⎢(𝑡2 − 𝑡2 ) ⋅ cos 2𝛼 1 𝑀∥ = ⎢ 𝑥2 𝑦2 2 ⎢ (𝑡𝑥 − 𝑡𝑦 ) ⋅ sin 2𝛼 ⎢ 0 ⎣
Optics Communications 413 (2018) 30–38
(𝑡2𝑥
(𝑡2𝑥 − 𝑡2𝑦 ) ⋅ cos 2𝛼 2 + 𝑡𝑦 ) ⋅ cos 2𝛼 + 2𝑡𝑥 𝑡𝑦 ⋅ sin 2𝛼 (𝑡𝑥 − 𝑡𝑦 )2 ⋅ sin 2𝛼 cos 2𝛼 0
(𝑡2𝑥 − 𝑡2𝑦 ) ⋅ sin 2𝛼 (𝑡𝑥 − 𝑡𝑦 )2 ⋅ sin 2𝛼 cos 2𝛼 2 (𝑡𝑥 + 𝑡2𝑦 ) ⋅ sin 2𝛼 + 2𝑡𝑥 𝑡𝑦 ⋅ cos 2𝛼 0
0 ⎤ 0 ⎥⎥ 0 ⎥ ⎥ 2𝑡𝑥 𝑡𝑦 ⎦
(15)
and ⎡ (𝑡2𝑥 + 𝑡2𝑦 ) ⎢−(𝑡2 − 𝑡2 ) ⋅ cos 2𝛼 1 𝑀⊥ = ⎢ 𝑥2 𝑦2 2 ⎢ −(𝑡𝑥 − 𝑡𝑦 ) ⋅ sin 2𝛼 ⎢ 0 ⎣
0 ⎤ 0 ⎥⎥ 0 ⎥ ⎥ 2𝑡𝑥 𝑡𝑦 ⎦
(16)
𝑺∥ (𝑖) = 𝑀∥ ⋅ 𝑺(𝑖) ⎡ ⎤ (𝑡2𝑥 + 𝑡2𝑦 ) ⋅ 𝑠0 (𝑖) + (𝑡2𝑥 − 𝑡2𝑦 ) cos 2𝛼 ⋅ 𝑠1 (𝑖) + (𝑡2𝑥 − 𝑡2𝑦 ) sin 2𝛼 ⋅ 𝑠2 (𝑖) ⎥ 1⎢ 2 2 = ⎢ (𝑡𝑥 − 𝑡𝑦 ) cos 2𝛼 ⋅ 𝑠0 (𝑖) + [(𝑡2𝑥 + 𝑡2𝑦 ) cos 2𝛼 + 2𝑡𝑥 𝑡𝑦 sin 2𝛼] ⋅ 𝑠1 (𝑖) + (𝑡𝑥 − 𝑡𝑦 )2 sin 2𝛼 cos 2𝛼 ⋅ 𝑠2 (𝑖) ⎥ 2⎢ ⎥ ⎣(𝑡2𝑥 − 𝑡2𝑦 ) sin 2𝛼 ⋅ 𝑠0 (𝑖) + (𝑡𝑥 − 𝑡𝑦 )2 sin 2𝛼 cos 2𝛼 ⋅ 𝑠1 (𝑖) + [(𝑡2𝑥 + 𝑡2𝑦 ) sin 2𝛼 + 2𝑡𝑥 𝑡𝑦 cos 2𝛼] ⋅ 𝑠2 (𝑖)0⎦
(17)
(𝑡2𝑥
−(𝑡2𝑥 − 𝑡2𝑦 ) ⋅ cos 2𝛼 2 + 𝑡𝑦 ) ⋅ cos 2𝛼 + 2𝑡𝑥 𝑡𝑦 ⋅ sin 2𝛼 (𝑡𝑥 − 𝑡𝑦 )2 ⋅ sin 2𝛼 cos 2𝛼 0
−(𝑡2𝑥 − 𝑡2𝑦 ) ⋅ sin 2𝛼 (𝑡𝑥 − 𝑡𝑦 )2 ⋅ sin 2𝛼 cos 2𝛼 2 (𝑡𝑥 + 𝑡2𝑦 ) ⋅ sin 2𝛼 + 2𝑡𝑥 𝑡𝑦 ⋅ cos 2𝛼 0
respectively. Box I.
and 𝑺⊥ (𝑖) = 𝑀⊥ ⋅ 𝑺(𝑖) ⎡ ⎤ (𝑡2𝑥 + 𝑡2𝑦 ) ⋅ 𝑠0 (𝑖) − (𝑡2𝑥 − 𝑡2𝑦 ) cos 2𝛼 ⋅ 𝑠1 (𝑖) − (𝑡2𝑥 − 𝑡2𝑦 ) sin 2𝛼 ⋅ 𝑠2 (𝑖) ⎥ 1⎢ 2 2 2 2 2 = ⎢ −(𝑡𝑥 − 𝑡𝑦 ) cos 2𝛼 ⋅ 𝑠0 (𝑖) + [(𝑡𝑥 + 𝑡𝑦 ) cos 2𝛼 + 2𝑡𝑥 𝑡𝑦 sin 2𝛼] ⋅ 𝑠1 (𝑖) + (𝑡𝑥 − 𝑡𝑦 ) sin 2𝛼 cos 2𝛼 ⋅ 𝑠2 (𝑖) ⎥ 2⎢ ⎥ ⎣−(𝑡2𝑥 − 𝑡2𝑦 ) sin 2𝛼 ⋅ 𝑠0 (𝑖) + (𝑡𝑥 − 𝑡𝑦 )2 sin 2𝛼 cos 2𝛼 ⋅ 𝑠1 (𝑖) + [(𝑡2𝑥 + 𝑡2𝑦 ) sin 2𝛼 + 2𝑡𝑥 𝑡𝑦 cos 2𝛼] ⋅ 𝑠2 (𝑖)0⎦
(18)
respectively, where 𝑖 indicates the scatter noise or the target signal. Box II.
and the target signal is obtained according to Eq. (10) or (20) [ ] ′ 𝐼PD (𝐷) = (𝑡2𝑥 − 𝑡2𝑦 ) cos 2𝜙V ⋅ 𝑠2 (𝐷) − tan 2𝜙V ⋅ 𝑠1 (𝐷)
(23)
where 𝜙V is the polarization angle of scatter noise. 4. Simulation The principle of PDI technique in a scattering environment has been investigated in Section 3. The common-mode rejection works relying on the angle of polarization of light. We further demonstrate the principle of PDI by simulation. Fig. 5 shows a pair of images consisting of scattering noise and target signal at different turbidity levels, which originates from [28], and is reset in this paper. In the left side of Fig. 5, polarization sum images are presented, which are equivalent to direct images. Decomposed images are shown in the right side of Fig. 5, in which the upper and lower sides represent the polarization images of the target signal and the scatter noise, respectively. The target used in the simulation is a combination of optical disk with rough surface and two aluminum sheets with smooth surface, the reflective character of which is similar. Here, we assume images meet the following three conditions: (a) the scatter noise and the target signal in the image have uniform distributions of polarization directions, respectively; (b) light from the aluminum sheets and the optical disk are partial polarized and un-polarized, respectively; (c) the polarization state analyzer is ideal, and the values of 𝑡𝑥 and 𝑡𝑦 are set to be one and zero, respectively, which has no effects on the evaluation of image quality. Fig. 5(a) and (b) correspond to the medium with low level of turbidity. In Fig. 5(a), the polarization directions of scatter noise and target signal are identical, both of which are 30◦ respect to the horizontal axis. In Fig. 5(b), there exists difference in the angle of polarization between the scatter noise and the target signal, which are 30◦ and 26.1◦ respect to the horizontal
Fig. 4. Numerical properties of function 𝑟(𝜃, 𝛿). (a) Three dimensional surface. (b) 𝛿 = 0. (c) 𝛿 = 𝜋∕4. (d) 𝛿 = 𝜋∕3.
𝛿 does not equal to zero. Thus PDI technique cannot improve image quality at all when 𝛿 is equal to zero. According to the above analysis, we can know that the principle of PDI is based on the difference in angle of polarization information between the scatter noise and target signal. Thus, when the angle 𝜃 is equal to 𝜋∕4, the scatter noise is eliminated, 33
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Fig. 5. Simulation of image scene. Low level of turbidity (a, b). High level of turbidity (c, d). 𝐼PS : polarization sum image.
axis, respectively. Fig. 5(c) and (d) correspond to the medium with high level of turbidity. The distributions of polarization directions for the scatter noise and the target signal are the combination of 23.6◦ and 32◦ (Fig. 5(c)), and that of 27◦ and 32◦ (Fig. 5(d)), respectively, referring to the horizontal axis. Fig. 6 shows common-mode rejection performance as a function of choice of polarization axis. Here, image quality is evaluated by the parameter of contrast, which is calculated by the formula 𝐶𝑜𝑛𝑡𝑟𝑎𝑠𝑡 = (𝐼max −𝐼min )∕(𝐼max +𝐼min ), where 𝐼max and 𝐼min represent the average light intensities corresponding to the regions of aluminum sheet and optical disk in the image, respectively. Fig. 6(a) and (b) show the cases at low level of turbidity, in which 𝜃 ranges from 0◦ to 55◦ in intervals of 5◦ . From Fig. 6(a), we can observe that the contrast is independent on the choice of polarization axis, and have a constant value of 0.0644. This can be explained that the scatter noise and the target signal have the same angle of polarization, and common-mode rejection is not valid, which has been demonstrated in Fig. 4(b). From Fig. 6(b), image contrast is closely related to the choice of polarization axis. When the orientation of 𝑃 𝑆𝐴1 is 75◦ , the image contrast has the optimum value of 0.4507. This is because the angle between orientations of scatter noise and 𝑃 𝑆𝐴1 is 45◦ , and scattering effect is eliminated by common-mode rejection. Fig. 6(c) and (d) show the cases at high level of turbidity, in which the values of 𝜃 are the same as that in Fig. 6(a). In both Fig. 6(c) and (d), the imaging performance is as a function of the choice of polarization axis because there exists difference in angle of polarization between the target signal and the scatter noise. And the image contrasts get the maximum values of 0.1837 (Fig. 6(c)) and 0.1587 (Fig. 6(d)), respectively, when the angle between the orientations of scatter noise and 𝑃 𝑆𝐴1 is 45◦ , in which the scatter noise is removed completely. To occupy the full display range, the pixel intensities in the image need to be transformed [18,23], shown as follows 𝐼(𝑥, 𝑦)trans = 𝜅[𝐼(𝑥, 𝑦)max − 𝐼(𝑥, 𝑦)],
of polarization axis is a time-consuming procedure, and polarization properties of the detected light may change over time, which affects the performance of PDI. Fig. 7 shows polarization filtering of scatter noise by traditional PDI technique with time dependent polarizations. Fig. 7(a) and (b) represent two orthogonal polarizations of scatter noise in high level of turbidity at 68.6◦ and −21.4◦ referring to Fig. 5(b). Fig. 7(c) shows the PDI component of scatter noise by means of commonmode rejection. In order to further describe the difference between these two orthogonal polarizations, histograms of intensity distributions corresponding to the images are given in the right side of Fig. 7. It can be observed that from the right side of Fig. 7(c), scatter noise could not be completely removed due to the varied polarization properties of light field over time. In fact, Brownian motion of particles in the medium happens all the time, which could affect the energy distributions of scattered light. Fig. 8 further shows another problem existed in the traditional PDI technique. The orientations of polarization axis meeting common-mode rejection are −21.4◦ and 68.6◦ , respectively, referring to Fig. 5(b). Assuming the minimum scale of polarization state analyzer’s dial in the simulation is one degree, the polarization axis could only take the proximate values of −21◦ and 69◦ , respectively, as shown in Fig. 8(a) and (b). From the left side of Fig. 8(a) and (b), these two orthogonal polarization images seems to be the same as each other. However, their intensity distributions are not exactly identical according to the histograms shown in the right side of Fig. 8(a) and (b). Thus, commonmode rejection fails in eliminating the scattering effect completely at this condition, as shown in Fig. 8(c). It should be noted that although the intensity of PDI component of the scatter noise is small, it still seriously degrades the image quality because the target signal attenuates rapidly at high level of turbidity. Figs. 7 and 8 point out the problems caused by mechanical rotation of polarizer in traditional PDI system. The key step of the solution is to obtain the polarization components of detected light instantaneously. We have known that PDI plays a role in descattering based on the polarization direction information. What is interesting is that the elements of Stokes vector 𝑠1 and 𝑠2 are relevant to the polarization direction of light field based on Eq. (3). According to Eqs. (19) and (20), PDI component can be expressed as a linear combination of the Stokes vector elements. More importantly, researches on the instruments of imaging polarimeter develop rapidly, which can acquire the Stokes vector of light instantaneously and precisely [29,30]. Therefore, it provides potential of using Stokes vector instead of mechanical rotation of polarization axis to achieve common-mode rejection for the purpose of effective PDI technique. However, it has been beyond the scope of this paper, which will be fully studied in the future.
(24)
where 𝐼(𝑥, 𝑦) represent the original intensity distributions of the image, 𝜅 is the scale factor, and is expressed as 𝜅 = 255∕[𝐼(𝑥, 𝑦)max − 𝐼(𝑥, 𝑦)min ],
(25)
where 𝐼(𝑥, 𝑦)max and 𝐼(𝑥, 𝑦)min are the maximum and the minimum values of pixel intensities in the image. The transformed images are shown in the insets of Fig. 6, in which the first and second ones of every pair of images represent the original and the re-scaled ones. We can directly see that the PDI technique could enhance the description of object features that invisible to direct imaging (Fig. 5(c) and (d)) in a scattering medium, which has the same effect as that in [18]. The results in Fig. 6 demonstrate that backscatter can be eliminated by common-mode rejection in PDI. However, mechanical rotation 34
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Fig. 6. PDI performance for low level of turbidity (a, b) and high level of turbidity (c, d).
5. Experiment
experimental evidence for detecting targets in turbid media based on the principle of PDI technology. Fig. 11 shows the typical images for the target hidden in turbid media with optical depth of 3.46 under different combinations of orthogonal polarizations. Here, the optical depth is calculated by the formula 𝑂𝐷 = 𝜇s ⋅ 𝐿, where 𝜇s is the scattering coefficient of media, and 𝐿 is distance from the target plane to the wall of container. From Fig. 11(a) to (f), the angles between the polarization axis orientation of 𝑃 𝑆𝐴1 and the polarization direction of scatter noise are 31◦ , 43◦ , 49◦ , 53◦ , 55◦ , and 61◦ , respectively. Fig. 11(g) further provides numerical profiles along the dotted line in Fig. 11(a) corresponding to different conditions. The imaging performance is as a function of the choice of polarization axis, which is demonstrated by the presentation of difference between the peak (aluminum sheet) and valley (wooden board) in Fig. 11(g). The most and least obvious differences can be observed at the conditions of 𝜃 = 43◦ and 𝜃 = 49◦ , respectively. In the former, the scatter noise is removed to a great extent, and the target signal is preserved. But in the latter, an opposite phenomenon occurs, which is determined by the choice of polarization axis. In order to investigate the performance of PDI system systematically, we change the scattering events for six different optical depths: 2.37, 2.81, 3.46, 4.11, 4.69, and 5.38. Fig. 12 shows a comparison of images obtained by direct imaging and PDI at different scattering conditions. For the PDI case, image with the optimum quality among different polarization axis choices is selected. Here, the imaging performance is described quantitatively by the parameter of contrast. And it is defined by the following formula 𝐶𝑜𝑛𝑡𝑟𝑎𝑠𝑡 = (𝐼max −𝐼min )∕(𝐼max +𝐼min ), where 𝐼max and 𝐼min are the average light intensities corresponding to the aluminum sheet region and wooden board region in the image, respectively. We can see from Fig. 12(a) that, the image contrasts for direct imaging decrease from 0.4413 to 0.1033 sharply with varying the optical depth of target from 2.37 to 5.38. This is because target signal received by the detector attenuates seriously and scattering events increase with increasing the optical depth of target in the medium. We can see from
5.1. Experimental setup Fig. 9 shows the schematic arrangement for verifying the simulation results. Tyo ever demonstrated that compared with passive PDI technology, active imaging could be performed in more highly scattering media [21]. Accordingly, a semiconductor laser working at the wavelength of 532 nm is used as the light source. A combination of beam expander and polarization state generator provides polarized incidence with beam size of 23 mm to illuminate the milk solution, which is contained in a 50 mm × 50 mm × 50 mm quartz cuvette and serves as the turbid medium to simulate a scattering environment. The target suspended in the medium consists of two parts: one is a pair of 1-cm2 aluminum sheets with smooth surface, the other is a wooden board with rough surface being Lambertian approximately, in which the former is pasted on the latter. The micro-displacement platform is used to adjust the depth of target in the medium. The images are recorded by an 8-bit CCD camera, in front of which a polarization state analyzer ensures polarization detection. In order to avoid saturation of the camera, a diameter-tunable aperture is applied to control the amounts of light entering the imaging system. 5.2. Results and discussion We firstly separate the scatter noise (aluminum sheet) and target signal (wooden board) based on the absorption method to describe the difference in polarization property between them. Fig. 10 gives the distributions of polarization direction for scatter noise and target signal at different optical depths. Here, the angle of polarization 𝜙 (Eq. (3)) is used to present the polarization direction information of light. It can be observed from Fig. 10(a) to (f) that, there exists difference in polarization direction between the scatter noise and the target signal at different scattering conditions. The results shown in Fig. 10 provide the 35
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Fig. 8. Traditional polarization filtering in PDI when the minimum scale of analyzer’s dial is not sufficient in descattering. (a) Polarization image corresponding to 𝑃 𝑆𝐴1 . (b) Polarization image corresponding to 𝑃 𝑆𝐴2 . (c) PDI component.
Fig. 7. Traditional polarization filtering in PDI when polarization properties of the scatter noise varies over time. (a) Polarization image corresponding to 𝑃 𝑆𝐴1 . (b) Polarization image corresponding to 𝑃 𝑆𝐴2 . (c) PDI component.
Fig. 12(b) that, the contrasts for PDI decrease from 0.6586 to 0.2365 slightly with increasing the optical depth of target. Comparing Fig. 12(a) with (b), a better image contrast can be obtained for PDI than that for direct imaging due to eliminating the scattering effect on the image. When the optical depth of target is 5.38, it is hard to discriminate the aluminum sheet from the wooden board for direct imaging. However, this discrimination could still be observed for PDI. 6. Summary In order to improve the efficiency of PDI in a scattering medium, we investigate the principle of PDI technique by means of Marius’s law and Mueller–Stokes formalism in this paper. PDI works based on the difference in polarization direction between the scatter noise and the target signal. When components of scatter noise at the orthogonal polarization axes of polarization state analyzer equal to each other, scattering effect is eliminated by use of common-mode rejection. It should be noted that if the scattering photons can be reused in the imaging configuration, the image quality can be enhanced further. Guo et al. ever proposed polarization retrieve method to reduce the scattering effect on the depolarization of polarized light through turbid media [31–33]. In the future work, we will introduce the polarization retrieve method into the PDI system to obtain better imaging performance.
Fig. 9. Experimental setup for PDI in turbid conditions. PSG: polarization state generator, MDP: micro-displacement platform, PSA: polarization state analyzer, A: aperture, L: imaging lens.
Eqs. (19) and (20) can be further sorted out as follows [ ] 𝐼PD (𝑖) = (𝑡2𝑥 − 𝑡2𝑦 ) cos 2𝛼 ⋅ 𝑠1 (𝑖) + sin 2𝛼 ⋅ 𝑠2 (𝑖) √ = (𝑡2𝑥 − 𝑡2𝑦 ) 𝑠21 (𝑖) + 𝑠22 (𝑖) √ √ √ ⎛√ ⎞ √ √ 𝑠21 (𝑖) 𝑠22 (𝑖) √ ⎜ × ⋅ cos 2𝛼 + √ ⋅ sin 2𝛼 ⎟ ⎜ 𝑠2 (𝑖) + 𝑠2 (𝑖) ⎟ 𝑠21 (𝑖) + 𝑠22 (𝑖) 1 2 ⎝ ⎠
(A.1)
where 𝑖 represents the component of scatter noise or target signal. On the basis of Eq. (3), Eq. (A.1) is changed into another formula √ [ ] 𝐼PD (𝑖) = (𝑡2𝑥 − 𝑡2𝑦 ) 𝑠21 (𝑖) + 𝑠22 (𝑖) cos 2𝛼𝑖 ⋅ cos 2𝛼 + sin 2𝛼𝑖 ⋅ sin 2𝛼 √ (A.2) = (𝑡2𝑥 − 𝑡2𝑦 ) 𝑠21 (𝑖) + 𝑠22 (𝑖) cos 2(𝛼 − 𝛼𝑖 )
Appendix We now show that PDI components of scatter noise and target signal obtained by Marius’s law and Mueller–Stokes formalism are identical with each other. 36
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Fig. 10. The distributions of angle of polarization for scatter noise and target signal at (a) 𝑂𝐷 = 2.37, (b) 𝑂𝐷 = 2.81, (c) 𝑂𝐷 = 3.46, (d) 𝑂𝐷 = 4.11, (e) 𝑂𝐷 = 4.69, (f) 𝑂𝐷 = 5.38.
Fig. 11. Image quality of PDI depends on the choice of polarization axis. (a) 𝜃 = 31◦ . (b) 𝜃 = 43◦ . (c) 𝜃 = 49◦ . (d) 𝜃 = 53◦ . (e) 𝜃 = 55◦ . (f) 𝜃 = 61◦ . (g) Numerical plots.
Fig. 12. Comparison of image contrasts using (a) direct imaging and (b) PDI. (C: image contrast).
where 𝛼𝑖 is the vector direction of scatter noise or target signal. According to geometric relationships in Figs. 2 and 3, the expression 𝛼 − 𝛼𝑖 is the angle between the direction of scatter noise or target signal and the polarization axis of polarization state analyzer. Comparing Eqs. (A.2), (9) and (10), the pair of Eqs. (9) and (19) or the pair of Eqs. (10) and (20) has identical expression. In this appendix, derivation is performed at the case of both linear elements of Stokes vector 𝑠1 and 𝑠2 being more than zero. It should be noted that Eq. (A.2) is still correct for other values of 𝑠1 and 𝑠2 , which is not repeated in this section.
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Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Investigation on principle of polarization-difference imaging in turbid conditions Wei Ren, Jinge Guan * School of Information and Communication Engineering, North University of China, Taiyuan 030051, China
a r t i c l e
i n f o
Keywords: Turbid media Polarimetric imaging Optical information processing
a b s t r a c t We investigate the principle of polarization-difference imaging (PDI) of objects in optically scattering environments. The work is performed by both Marius’s law and Mueller–Stokes formalism, and is further demonstrated by simulation. The results show that the object image is obtained based on the difference in polarization direction between the scatter noise and the target signal, and imaging performance is closely related to the choice of polarization analyzer axis. In addition, this study illustrates the potential of Stoke vector for promoting application of PDI system in the real world scene. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Optical imaging through turbid media has important applications in the fields of medical diagnosis, underwater target detection, and driving in the fog. And, it is challenging because light scattering and absorption effects degrade image quality [1–3]. When interactions between the light and the medium are dominated by absorption, imaging performance can be improved by enhancing the illumination energy or the detector sensitivity. However, problems caused by scattering are more serious due to scattered light is superimposed on the target signal to be received by the detector, which is responsible for image blurring and contrast reduction. Image processing and physical model based detection are the two current descattering methods to enhance image quality [4,5]. Compared with the latter, the former is not sufficient to remove scattering effect from images [6]. When using physical model based method to detect and recognize objects in turbid conditions, the principle is based on the difference in physical properties, such as timeof-flight [7,8], polarization [9,10], and coherence [11], between the scatter noise and the target signal. Among the above imaging methods, polarimetric imaging through turbid media is promising due to its advantage of low cost and easy operation. Also, it has attracted more and more attention of researchers because certain problems still need to be solved for the purpose of application in the real world. The research progresses in active polarimetric imaging aspect are mainly as follows. Under polarization illumination, polarization imaging with a cross polarization detection could increase the visibility range in scattering media compared with
direct imaging [12]. In the cross polarization detection method, circular polarization shows a better performance than linear polarization due to the former is depolarized by the object more than the latter [13]. Investigations on the interactions of object and polarized light by Mueller matrix further demonstrate that circular polarization is more depolarized than linear polarization [14]. Asymmetric segmented phaseonly filter is introduced to cross polarization detection method to enhance visibility [15]. Co-polarization image is further acquired, and is combined with cross polarization image to recover the depolarized object in a scattering medium [10]. Optical correlation-based approach is used to improve performance of polarization recovery [16]. And polarization recovery method by estimating the polarized-difference signal can obtain not only depolarized but also polarized objects [17]. For active detection method, the polarization state of light has been known, and is typically under the experimenters’ control [18]. It should be noted that polarization detection with passive mode is seldom concerned. Polarization-difference imaging (PDI) is a bio-inspired technique that uncovers the features of object surface in a scattering environment with ambient light [18]. Quantitative research shows that when compared with direct imaging, PDI increases distance at which certain target features can be detected by 2–3 times [19]. And a combination of direct imaging and PDI is suitable for detecting weakly polarized objects in scattering media, which could be well characterized by colorimetric representations [20]. PDI is further demonstrated to enhance the pointspread function for imaging [21]. Adaptive algorithms are proposed to
* Corresponding author.
E-mail address: [email protected] (J. Guan). https://doi.org/10.1016/j.optcom.2017.12.025 Received 16 September 2017; Received in revised form 7 November 2017; Accepted 10 December 2017 0030-4018/© 2017 Elsevier B.V. All rights reserved.
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Fig. 1. Imaging model in the scattering medium: (1) direction transmission, (2) veiling light, (3) light lost due to scattering, (4) light absorbed by the medium.
Fig. 2. Polarization filtering in PDI. 𝑉 : veiling light, 𝐷: direct transmission. 𝑃 𝑆𝐴1 and 𝑃 𝑆𝐴2 are the orthogonal polarizations, respectively.
deal with diversity of polarization direction distributions [22]. Micropolarizer array is used to enhance image contrast for PDI technique by controlling a micropatterned liquid-crystal cell alignment [23]. However, for traditional PDI systems, common-mode rejection of scattering effect is performed by mechanical rotation of the axis of polarization state analyzer, which is helpless in implementing real-time detection. In fact, polarimetric imaging in a scattering medium is based on the difference in polarization responses between the target signal and the scatter noise. In order to improve the performance of PDI technique, the principle of PDI should be understood. This is exactly what we want to study in the paper.
the difference between 45◦ and 135◦ polarizations, 𝑠3 represents the difference between right and left circular polarizations, 𝑇 is the symbol of transpose matrix. Two polarization parameters, i.e. degree of polarization (𝐷𝑜𝑃 ) and angle of polarization (𝜙), can be obtained from the Stokes vector √ 𝑠21 + 𝑠22 + 𝑠23 𝐷𝑜𝑃 = (2) 𝑠0 and 𝜙=
2. Imaging model
(3)
respectively. Degree of polarization indicates the proportion of polarized light in the total light intensity, and angle of polarization represents the polarization direction of polarized light. Light can be classified into completely polarized light (𝐷𝑜𝑃 = 1), un-polarized light (𝐷𝑜𝑃 = 0), and partially polarized light (0 < 𝐷𝑜𝑃 < 1). Laan et al. ever considered the partially polarized light as the sum of completely polarized light and unpolarized light [27]. Here, we further decompose partial polarized light into linearly polarized, circularly polarized, and un-polarized light, as shown in Eq. (4)
For passive optical imaging in a scattering medium with ambient illumination, light undergoing scattering events from the target can be classified into four categories [19,24], as shown in Fig. 1. The direction transmission is the component corresponding to the target. Veiling light comes from particles in front of the target, which is also referred as path radiance [25]. Light being scattered away from the detector and that being absorbed by the medium could not be recorded. Thus we only consider the components of direction transmission and veiling light. The latter contains no target information and causes the degradation of image quality. In this paper, we investigate the principle of PDI based on the difference in polarization properties between the direction transmission and the veiling light.
𝑺 = 𝑺linearly−polarized + 𝑺circularly−polarized + 𝑺un−polarized √ √ ⎡ 𝑠2 + 𝑠2 ⎤ ⎡ ⎤ ⎡𝑠 − 𝑠2 + 𝑠2 − 𝑠 ⎤ 𝑠 3⎥ 0 3 1 2 1 2 ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ 0 𝑠1 0 ⎥+⎢ ⎥+⎢ ⎥. = ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ 0 𝑠 0 2 ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ 0 0 ⎦ ⎣ 𝑠3 ⎦ ⎣ ⎣ ⎦
3. Principle
(4)
On the basis of polarization theory, for both circularly polarized and un-polarized light, their decompositions in arbitrary orthogonal polarization directions equal to each other. Referring to Eq. (1), circular polarization and un-polarized components of the detected light become zero in the process of PDI. Therefore, what the PDI technique actually deals with is the linear polarization component. Fig. 2 gives the physical process of polarization filtering for PDI. We suppose that the angle between the orientation of one polarization axis of the polarization state analyzer (𝑃 𝑆𝐴1 ) and the polarization direction of scatter noise is 𝜃, and the angle between polarization directions of target signal and scatter noise is 𝛿, respectively. According to Eq. (4), √
3.1. Concept of PDI Motivated by the visual system of green sunfish which is sensitive to polarization for serving in imaging [26], two orthogonal polarization images are captured, and the concept of bio-inspired PDI is defined by calculating the difference between these two images [18] 𝐼PD (𝑥, 𝑦) = 𝐼∥ (𝑥, 𝑦) − 𝐼⊥ (𝑥, 𝑦)
𝑠 1 arctan( 2 ) 2 𝑠1
(1)
where symbols of ∥ and ⊥ represent two arbitrary orthogonal polarizations, and (𝑥, 𝑦) indicates the pixel position in the image. In the optical polarization information processing (Eq. (1)), imaging performance is dependent on the choice of polarization axis, and the scatter noise is removed by means of common-mode rejection. Under what circumstances can this descattering method work? With such an object, we further investigate the physical process of this polarization filtering based on the Marius’s law and the Mueller–Stokes formalism.
the linear polarization intensity of light is 𝑠21 + 𝑠22 . And polarization components in the orthogonal polarization directions can be obtained based on Marius’s law. For the scatter noise, its polarization components in the directions of 𝑃 𝑆𝐴1 and 𝑃 𝑆𝐴2 are √ 𝐼∥ (𝑉 ) = 𝑠21 (𝑉 ) + 𝑠22 (𝑉 )(𝑡2𝑥 ⋅ cos 𝜃 + 𝑡2𝑦 ⋅ sin 𝜃) (5)
3.2. Marius’s law based analysis
and Stokes vector, which consists of four elements, is commonly used to describe the polarization state of light. This vector can be expressed by the formula S = [𝑠0 , 𝑠1 , 𝑠2 , 𝑠3 ]T , where 𝑠0 is the total light intensity, 𝑠1 indicates the difference between 0◦ and 90◦ polarizations, 𝑠2 is
𝐼⊥ (𝑉 ) =
√
𝑠21 (𝑉 ) + 𝑠22 (𝑉 )(𝑡2𝑥 ⋅ sin 𝜃 + 𝑡2𝑦 ⋅ cos 𝜃)
(6)
respectively, where 𝑡𝑥 and 𝑡𝑦 are the maximum and minimum transmission coefficients of polarization state analyzer, respectively. 31
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Optics Communications 413 (2018) 30–38
For the target signal, its polarization components in the directions of 𝑃 𝑆𝐴1 and 𝑃 𝑆𝐴2 are √ [ ] (7) 𝐼∥ (𝐷) = 𝑠21 (𝐷) + 𝑠22 (𝐷) 𝑡2𝑥 ⋅ cos 2(𝜃 + 𝛿) + 𝑡2𝑦 ⋅ sin 2(𝜃 + 𝛿) and √ 𝐼⊥ (𝐷) =
𝑠21 (𝐷) + 𝑠22 (𝐷)[𝑡2𝑥 ⋅ sin 2(𝜃 + 𝛿) + 𝑡2𝑦 ⋅ cos 2(𝜃 + 𝛿)]
(8)
respectively. On the basis of Eq. (1), the PDI components of scatter noise and target signal are 𝐼PD (𝑉 ) = 𝐼∥ (𝑉 ) − 𝐼⊥ (𝑉 ) √ = (𝑡2𝑥 − 𝑡2𝑦 ) ⋅ 𝑠21 (𝑉 ) + 𝑠22 (𝑉 ) cos 2𝜃
Fig. 3. Orientation distributions of polarization axis. (a) 0 ≤ 𝛼 ≤ 𝜋∕2. (b) −𝜋∕2 ≤ 𝛼 > 0.
(9)
On the basis of Eqs. (15) and (16), the polarization states of detected light on the two orthogonal polarization directions are expressed as equations in Box II. Since the first element of Stokes vector represents the total intensity of light, the PDI components of scatter noise and target signal are calculated referring to Eqs. (17) and (18), which are shown as
and 𝐼PD (𝐷) = 𝐼∥ (𝐷) − 𝐼⊥ (𝐷) √ = (𝑡2𝑥 − 𝑡2𝑦 ) ⋅ 𝑠21 (𝐷) + 𝑠22 (𝐷) cos 2(𝜃 + 𝛿)
(10)
respectively.
𝐼PD (𝑉 ) = 𝐼∥ (𝑉 ) − 𝐼⊥ (𝑉 ) = (𝑡2𝑥 − 𝑡2𝑦 )[cos 2𝛼 ⋅ 𝑠1 (𝑉 ) + sin 2𝛼 ⋅ 𝑠2 (𝑉 )]
3.3. Mueller–Stokes formalism based analysis
and 𝐼PD (𝐷) = 𝐼∥ (𝐷) − 𝐼⊥ (𝐷)
When light interacts with the object, the relationship between the polarization state of transmitted light and that of incident light can be expressed as
= (𝑡2𝑥 − 𝑡2𝑦 )[cos 2𝛼 ⋅ 𝑠1 (𝐷) + sin 2𝛼 ⋅ 𝑠2 (𝐷)]
where 𝑺out and 𝑺in are Stokes vectors of transmitted light and incident light, respectively, 𝑴 represent the Mueller matrix of the object, which is a 4 × 4 array. On the basis of optical polarization theory, Mueller matrix of the polarization state analyzer with polarization axis orientated at the angle of 𝛼 with respect to the horizontal direction is calculated as 𝑀PSA (𝛼) = 𝐴(−𝛼)𝑀PSA (0◦ )𝐴(𝛼)
3.4. PDI quality evaluation Since the performance of PDI depends on the polarization axis of polarization state analyzer [18], we use the ratio of PDI components of scatter noise and target signal in Sections 3.2 and 3.3 to investigate how the choice of polarization axis affects the image quality, which is expressed as
(12)
where 𝐴(𝛼) corresponds to the 4×4 rotation matrix with 𝛼 varying from −𝜋∕2 to 𝜋/2, and 𝑀𝑃 𝑆𝐴 (0◦ ) is the Mueller matrix of polarization state analyzer with polarization axis orientated at the angle of 0◦ . The formulas of them can be expressed as ⎡1 ⎢ 0 𝐴(𝛼) = ⎢ ⎢0 ⎢0 ⎣
0 cos 2𝛼 − sin 2𝛼 0
0 sin 2𝛼 cos 2𝛼 0
0⎤ ⎥ 0⎥ 0⎥ 1⎥⎦
⎡𝑡2𝑥 + 𝑡2𝑦 ⎢2 𝑡 − 𝑡2𝑦 𝑀PSA (0◦ ) = ⎢ 𝑥 ⎢ 0 ⎢ ⎣ 0
𝑡2𝑥 − 𝑡2𝑦 𝑡2𝑥 + 𝑡2𝑦 0 0
0 0 2𝑡𝑥 𝑡𝑦 0
𝐼PD (𝑉 ) 𝐼PD (𝐷) √ 𝑠21 (𝑉 ) + 𝑠22 (𝑉 ) = √ ⋅ 𝑟(𝜃, 𝛿) 𝑠21 (𝐷) + 𝑠22 (𝐷)
𝑅(𝜃, 𝛿) =
(21)
where the formula of 𝑟(𝜃, 𝛿) is
(13)
𝑟(𝜃, 𝛿) =
and 0 ⎤ ⎥ 0 ⎥ 0 ⎥ ⎥ 2𝑡𝑥 𝑡𝑦 ⎦
(20)
respectively. We have analyzed the physical process of PDI by Marius’s law and Mueller–Stokes formalism, and the contents described by them are identical in essence, which is provided in Appendix
(11)
𝑺out = 𝑀 ⋅ 𝑺in
(19)
cos 2𝜃 . cos 2(𝜃 + 𝛿)
(22)
It can be observed from Eq. (21) that, image quality obtained by the PDI technique is proportional to the value of 𝑅(𝜃, 𝛿). Fig. 4 further gives numerical plots corresponding to the function 𝑟(𝜃, 𝛿), in which both 𝜃 and 𝛿 range from 0 to 𝜋/2. For the purpose of simplified analysis, linear polarization components of 𝑠1 and 𝑠2 are neglected due to they are fixed in the image and not affected by the choice of polarization axis. In Fig. 4(a), the value of 𝑟(𝜃, 𝛿) is affected by the value of 𝜃 when 𝛿 does not equal to zero, and it is always zero with 𝜃 equaling to 𝜋/4. An important feature is that 𝑟(𝜃, 𝛿) always equal to one when 𝛿 equals zero, which illustrates that image quality obtained by PDI technique is nothing to do with the choice of polarization axis. In order to understand the properties of function 𝑟(𝜃, 𝛿) intuitively, two dimensional numerical plots at the case of 𝛿 equaling to 0, 𝜋∕4 and 𝜋∕3, subsequently, are further given in Fig. 4(b)–(d). These results demonstrate that when angles between the scatter noise and each of the orthogonal polarization axis are both 𝜋/4, the scattering effect can be eliminated by means of common-mode rejection. It should also be noted that, common-mode rejection based descattering method is valid only when the value of
(14)
respectively. When using the PDI technique to detect objects in a scattering environment, we suppose that the angle between the orientation of one polarization axis of the polarization state analyzer (𝑃 𝑆𝐴1 ) and the horizontal direction is arbitrary with a value of 𝛼. It can be observed from Fig. 3 that when computing the difference between these two orthogonal polarizations (𝑃 𝑆𝐴1 and 𝑃 𝑆𝐴2 ), results corresponding to Fig. 3(a) and (b) have opposite values. Thus only the case in Fig. 3(a) is analyzed in this paper. Combining Eqs. (12)–(14), Mueller matrices of the polarization state analyzer corresponding to 𝑃 𝑆𝐴1 and 𝑃 𝑆𝐴2 are calculated as equations given in Box I. 32
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⎡ (𝑡2𝑥 + 𝑡2𝑦 ) ⎢(𝑡2 − 𝑡2 ) ⋅ cos 2𝛼 1 𝑀∥ = ⎢ 𝑥2 𝑦2 2 ⎢ (𝑡𝑥 − 𝑡𝑦 ) ⋅ sin 2𝛼 ⎢ 0 ⎣
Optics Communications 413 (2018) 30–38
(𝑡2𝑥
(𝑡2𝑥 − 𝑡2𝑦 ) ⋅ cos 2𝛼 2 + 𝑡𝑦 ) ⋅ cos 2𝛼 + 2𝑡𝑥 𝑡𝑦 ⋅ sin 2𝛼 (𝑡𝑥 − 𝑡𝑦 )2 ⋅ sin 2𝛼 cos 2𝛼 0
(𝑡2𝑥 − 𝑡2𝑦 ) ⋅ sin 2𝛼 (𝑡𝑥 − 𝑡𝑦 )2 ⋅ sin 2𝛼 cos 2𝛼 2 (𝑡𝑥 + 𝑡2𝑦 ) ⋅ sin 2𝛼 + 2𝑡𝑥 𝑡𝑦 ⋅ cos 2𝛼 0
0 ⎤ 0 ⎥⎥ 0 ⎥ ⎥ 2𝑡𝑥 𝑡𝑦 ⎦
(15)
and ⎡ (𝑡2𝑥 + 𝑡2𝑦 ) ⎢−(𝑡2 − 𝑡2 ) ⋅ cos 2𝛼 1 𝑀⊥ = ⎢ 𝑥2 𝑦2 2 ⎢ −(𝑡𝑥 − 𝑡𝑦 ) ⋅ sin 2𝛼 ⎢ 0 ⎣
0 ⎤ 0 ⎥⎥ 0 ⎥ ⎥ 2𝑡𝑥 𝑡𝑦 ⎦
(16)
𝑺∥ (𝑖) = 𝑀∥ ⋅ 𝑺(𝑖) ⎡ ⎤ (𝑡2𝑥 + 𝑡2𝑦 ) ⋅ 𝑠0 (𝑖) + (𝑡2𝑥 − 𝑡2𝑦 ) cos 2𝛼 ⋅ 𝑠1 (𝑖) + (𝑡2𝑥 − 𝑡2𝑦 ) sin 2𝛼 ⋅ 𝑠2 (𝑖) ⎥ 1⎢ 2 2 = ⎢ (𝑡𝑥 − 𝑡𝑦 ) cos 2𝛼 ⋅ 𝑠0 (𝑖) + [(𝑡2𝑥 + 𝑡2𝑦 ) cos 2𝛼 + 2𝑡𝑥 𝑡𝑦 sin 2𝛼] ⋅ 𝑠1 (𝑖) + (𝑡𝑥 − 𝑡𝑦 )2 sin 2𝛼 cos 2𝛼 ⋅ 𝑠2 (𝑖) ⎥ 2⎢ ⎥ ⎣(𝑡2𝑥 − 𝑡2𝑦 ) sin 2𝛼 ⋅ 𝑠0 (𝑖) + (𝑡𝑥 − 𝑡𝑦 )2 sin 2𝛼 cos 2𝛼 ⋅ 𝑠1 (𝑖) + [(𝑡2𝑥 + 𝑡2𝑦 ) sin 2𝛼 + 2𝑡𝑥 𝑡𝑦 cos 2𝛼] ⋅ 𝑠2 (𝑖)0⎦
(17)
(𝑡2𝑥
−(𝑡2𝑥 − 𝑡2𝑦 ) ⋅ cos 2𝛼 2 + 𝑡𝑦 ) ⋅ cos 2𝛼 + 2𝑡𝑥 𝑡𝑦 ⋅ sin 2𝛼 (𝑡𝑥 − 𝑡𝑦 )2 ⋅ sin 2𝛼 cos 2𝛼 0
−(𝑡2𝑥 − 𝑡2𝑦 ) ⋅ sin 2𝛼 (𝑡𝑥 − 𝑡𝑦 )2 ⋅ sin 2𝛼 cos 2𝛼 2 (𝑡𝑥 + 𝑡2𝑦 ) ⋅ sin 2𝛼 + 2𝑡𝑥 𝑡𝑦 ⋅ cos 2𝛼 0
respectively. Box I.
and 𝑺⊥ (𝑖) = 𝑀⊥ ⋅ 𝑺(𝑖) ⎡ ⎤ (𝑡2𝑥 + 𝑡2𝑦 ) ⋅ 𝑠0 (𝑖) − (𝑡2𝑥 − 𝑡2𝑦 ) cos 2𝛼 ⋅ 𝑠1 (𝑖) − (𝑡2𝑥 − 𝑡2𝑦 ) sin 2𝛼 ⋅ 𝑠2 (𝑖) ⎥ 1⎢ 2 2 2 2 2 = ⎢ −(𝑡𝑥 − 𝑡𝑦 ) cos 2𝛼 ⋅ 𝑠0 (𝑖) + [(𝑡𝑥 + 𝑡𝑦 ) cos 2𝛼 + 2𝑡𝑥 𝑡𝑦 sin 2𝛼] ⋅ 𝑠1 (𝑖) + (𝑡𝑥 − 𝑡𝑦 ) sin 2𝛼 cos 2𝛼 ⋅ 𝑠2 (𝑖) ⎥ 2⎢ ⎥ ⎣−(𝑡2𝑥 − 𝑡2𝑦 ) sin 2𝛼 ⋅ 𝑠0 (𝑖) + (𝑡𝑥 − 𝑡𝑦 )2 sin 2𝛼 cos 2𝛼 ⋅ 𝑠1 (𝑖) + [(𝑡2𝑥 + 𝑡2𝑦 ) sin 2𝛼 + 2𝑡𝑥 𝑡𝑦 cos 2𝛼] ⋅ 𝑠2 (𝑖)0⎦
(18)
respectively, where 𝑖 indicates the scatter noise or the target signal. Box II.
and the target signal is obtained according to Eq. (10) or (20) [ ] ′ 𝐼PD (𝐷) = (𝑡2𝑥 − 𝑡2𝑦 ) cos 2𝜙V ⋅ 𝑠2 (𝐷) − tan 2𝜙V ⋅ 𝑠1 (𝐷)
(23)
where 𝜙V is the polarization angle of scatter noise. 4. Simulation The principle of PDI technique in a scattering environment has been investigated in Section 3. The common-mode rejection works relying on the angle of polarization of light. We further demonstrate the principle of PDI by simulation. Fig. 5 shows a pair of images consisting of scattering noise and target signal at different turbidity levels, which originates from [28], and is reset in this paper. In the left side of Fig. 5, polarization sum images are presented, which are equivalent to direct images. Decomposed images are shown in the right side of Fig. 5, in which the upper and lower sides represent the polarization images of the target signal and the scatter noise, respectively. The target used in the simulation is a combination of optical disk with rough surface and two aluminum sheets with smooth surface, the reflective character of which is similar. Here, we assume images meet the following three conditions: (a) the scatter noise and the target signal in the image have uniform distributions of polarization directions, respectively; (b) light from the aluminum sheets and the optical disk are partial polarized and un-polarized, respectively; (c) the polarization state analyzer is ideal, and the values of 𝑡𝑥 and 𝑡𝑦 are set to be one and zero, respectively, which has no effects on the evaluation of image quality. Fig. 5(a) and (b) correspond to the medium with low level of turbidity. In Fig. 5(a), the polarization directions of scatter noise and target signal are identical, both of which are 30◦ respect to the horizontal axis. In Fig. 5(b), there exists difference in the angle of polarization between the scatter noise and the target signal, which are 30◦ and 26.1◦ respect to the horizontal
Fig. 4. Numerical properties of function 𝑟(𝜃, 𝛿). (a) Three dimensional surface. (b) 𝛿 = 0. (c) 𝛿 = 𝜋∕4. (d) 𝛿 = 𝜋∕3.
𝛿 does not equal to zero. Thus PDI technique cannot improve image quality at all when 𝛿 is equal to zero. According to the above analysis, we can know that the principle of PDI is based on the difference in angle of polarization information between the scatter noise and target signal. Thus, when the angle 𝜃 is equal to 𝜋∕4, the scatter noise is eliminated, 33
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Fig. 5. Simulation of image scene. Low level of turbidity (a, b). High level of turbidity (c, d). 𝐼PS : polarization sum image.
axis, respectively. Fig. 5(c) and (d) correspond to the medium with high level of turbidity. The distributions of polarization directions for the scatter noise and the target signal are the combination of 23.6◦ and 32◦ (Fig. 5(c)), and that of 27◦ and 32◦ (Fig. 5(d)), respectively, referring to the horizontal axis. Fig. 6 shows common-mode rejection performance as a function of choice of polarization axis. Here, image quality is evaluated by the parameter of contrast, which is calculated by the formula 𝐶𝑜𝑛𝑡𝑟𝑎𝑠𝑡 = (𝐼max −𝐼min )∕(𝐼max +𝐼min ), where 𝐼max and 𝐼min represent the average light intensities corresponding to the regions of aluminum sheet and optical disk in the image, respectively. Fig. 6(a) and (b) show the cases at low level of turbidity, in which 𝜃 ranges from 0◦ to 55◦ in intervals of 5◦ . From Fig. 6(a), we can observe that the contrast is independent on the choice of polarization axis, and have a constant value of 0.0644. This can be explained that the scatter noise and the target signal have the same angle of polarization, and common-mode rejection is not valid, which has been demonstrated in Fig. 4(b). From Fig. 6(b), image contrast is closely related to the choice of polarization axis. When the orientation of 𝑃 𝑆𝐴1 is 75◦ , the image contrast has the optimum value of 0.4507. This is because the angle between orientations of scatter noise and 𝑃 𝑆𝐴1 is 45◦ , and scattering effect is eliminated by common-mode rejection. Fig. 6(c) and (d) show the cases at high level of turbidity, in which the values of 𝜃 are the same as that in Fig. 6(a). In both Fig. 6(c) and (d), the imaging performance is as a function of the choice of polarization axis because there exists difference in angle of polarization between the target signal and the scatter noise. And the image contrasts get the maximum values of 0.1837 (Fig. 6(c)) and 0.1587 (Fig. 6(d)), respectively, when the angle between the orientations of scatter noise and 𝑃 𝑆𝐴1 is 45◦ , in which the scatter noise is removed completely. To occupy the full display range, the pixel intensities in the image need to be transformed [18,23], shown as follows 𝐼(𝑥, 𝑦)trans = 𝜅[𝐼(𝑥, 𝑦)max − 𝐼(𝑥, 𝑦)],
of polarization axis is a time-consuming procedure, and polarization properties of the detected light may change over time, which affects the performance of PDI. Fig. 7 shows polarization filtering of scatter noise by traditional PDI technique with time dependent polarizations. Fig. 7(a) and (b) represent two orthogonal polarizations of scatter noise in high level of turbidity at 68.6◦ and −21.4◦ referring to Fig. 5(b). Fig. 7(c) shows the PDI component of scatter noise by means of commonmode rejection. In order to further describe the difference between these two orthogonal polarizations, histograms of intensity distributions corresponding to the images are given in the right side of Fig. 7. It can be observed that from the right side of Fig. 7(c), scatter noise could not be completely removed due to the varied polarization properties of light field over time. In fact, Brownian motion of particles in the medium happens all the time, which could affect the energy distributions of scattered light. Fig. 8 further shows another problem existed in the traditional PDI technique. The orientations of polarization axis meeting common-mode rejection are −21.4◦ and 68.6◦ , respectively, referring to Fig. 5(b). Assuming the minimum scale of polarization state analyzer’s dial in the simulation is one degree, the polarization axis could only take the proximate values of −21◦ and 69◦ , respectively, as shown in Fig. 8(a) and (b). From the left side of Fig. 8(a) and (b), these two orthogonal polarization images seems to be the same as each other. However, their intensity distributions are not exactly identical according to the histograms shown in the right side of Fig. 8(a) and (b). Thus, commonmode rejection fails in eliminating the scattering effect completely at this condition, as shown in Fig. 8(c). It should be noted that although the intensity of PDI component of the scatter noise is small, it still seriously degrades the image quality because the target signal attenuates rapidly at high level of turbidity. Figs. 7 and 8 point out the problems caused by mechanical rotation of polarizer in traditional PDI system. The key step of the solution is to obtain the polarization components of detected light instantaneously. We have known that PDI plays a role in descattering based on the polarization direction information. What is interesting is that the elements of Stokes vector 𝑠1 and 𝑠2 are relevant to the polarization direction of light field based on Eq. (3). According to Eqs. (19) and (20), PDI component can be expressed as a linear combination of the Stokes vector elements. More importantly, researches on the instruments of imaging polarimeter develop rapidly, which can acquire the Stokes vector of light instantaneously and precisely [29,30]. Therefore, it provides potential of using Stokes vector instead of mechanical rotation of polarization axis to achieve common-mode rejection for the purpose of effective PDI technique. However, it has been beyond the scope of this paper, which will be fully studied in the future.
(24)
where 𝐼(𝑥, 𝑦) represent the original intensity distributions of the image, 𝜅 is the scale factor, and is expressed as 𝜅 = 255∕[𝐼(𝑥, 𝑦)max − 𝐼(𝑥, 𝑦)min ],
(25)
where 𝐼(𝑥, 𝑦)max and 𝐼(𝑥, 𝑦)min are the maximum and the minimum values of pixel intensities in the image. The transformed images are shown in the insets of Fig. 6, in which the first and second ones of every pair of images represent the original and the re-scaled ones. We can directly see that the PDI technique could enhance the description of object features that invisible to direct imaging (Fig. 5(c) and (d)) in a scattering medium, which has the same effect as that in [18]. The results in Fig. 6 demonstrate that backscatter can be eliminated by common-mode rejection in PDI. However, mechanical rotation 34
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Fig. 6. PDI performance for low level of turbidity (a, b) and high level of turbidity (c, d).
5. Experiment
experimental evidence for detecting targets in turbid media based on the principle of PDI technology. Fig. 11 shows the typical images for the target hidden in turbid media with optical depth of 3.46 under different combinations of orthogonal polarizations. Here, the optical depth is calculated by the formula 𝑂𝐷 = 𝜇s ⋅ 𝐿, where 𝜇s is the scattering coefficient of media, and 𝐿 is distance from the target plane to the wall of container. From Fig. 11(a) to (f), the angles between the polarization axis orientation of 𝑃 𝑆𝐴1 and the polarization direction of scatter noise are 31◦ , 43◦ , 49◦ , 53◦ , 55◦ , and 61◦ , respectively. Fig. 11(g) further provides numerical profiles along the dotted line in Fig. 11(a) corresponding to different conditions. The imaging performance is as a function of the choice of polarization axis, which is demonstrated by the presentation of difference between the peak (aluminum sheet) and valley (wooden board) in Fig. 11(g). The most and least obvious differences can be observed at the conditions of 𝜃 = 43◦ and 𝜃 = 49◦ , respectively. In the former, the scatter noise is removed to a great extent, and the target signal is preserved. But in the latter, an opposite phenomenon occurs, which is determined by the choice of polarization axis. In order to investigate the performance of PDI system systematically, we change the scattering events for six different optical depths: 2.37, 2.81, 3.46, 4.11, 4.69, and 5.38. Fig. 12 shows a comparison of images obtained by direct imaging and PDI at different scattering conditions. For the PDI case, image with the optimum quality among different polarization axis choices is selected. Here, the imaging performance is described quantitatively by the parameter of contrast. And it is defined by the following formula 𝐶𝑜𝑛𝑡𝑟𝑎𝑠𝑡 = (𝐼max −𝐼min )∕(𝐼max +𝐼min ), where 𝐼max and 𝐼min are the average light intensities corresponding to the aluminum sheet region and wooden board region in the image, respectively. We can see from Fig. 12(a) that, the image contrasts for direct imaging decrease from 0.4413 to 0.1033 sharply with varying the optical depth of target from 2.37 to 5.38. This is because target signal received by the detector attenuates seriously and scattering events increase with increasing the optical depth of target in the medium. We can see from
5.1. Experimental setup Fig. 9 shows the schematic arrangement for verifying the simulation results. Tyo ever demonstrated that compared with passive PDI technology, active imaging could be performed in more highly scattering media [21]. Accordingly, a semiconductor laser working at the wavelength of 532 nm is used as the light source. A combination of beam expander and polarization state generator provides polarized incidence with beam size of 23 mm to illuminate the milk solution, which is contained in a 50 mm × 50 mm × 50 mm quartz cuvette and serves as the turbid medium to simulate a scattering environment. The target suspended in the medium consists of two parts: one is a pair of 1-cm2 aluminum sheets with smooth surface, the other is a wooden board with rough surface being Lambertian approximately, in which the former is pasted on the latter. The micro-displacement platform is used to adjust the depth of target in the medium. The images are recorded by an 8-bit CCD camera, in front of which a polarization state analyzer ensures polarization detection. In order to avoid saturation of the camera, a diameter-tunable aperture is applied to control the amounts of light entering the imaging system. 5.2. Results and discussion We firstly separate the scatter noise (aluminum sheet) and target signal (wooden board) based on the absorption method to describe the difference in polarization property between them. Fig. 10 gives the distributions of polarization direction for scatter noise and target signal at different optical depths. Here, the angle of polarization 𝜙 (Eq. (3)) is used to present the polarization direction information of light. It can be observed from Fig. 10(a) to (f) that, there exists difference in polarization direction between the scatter noise and the target signal at different scattering conditions. The results shown in Fig. 10 provide the 35
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Fig. 8. Traditional polarization filtering in PDI when the minimum scale of analyzer’s dial is not sufficient in descattering. (a) Polarization image corresponding to 𝑃 𝑆𝐴1 . (b) Polarization image corresponding to 𝑃 𝑆𝐴2 . (c) PDI component.
Fig. 7. Traditional polarization filtering in PDI when polarization properties of the scatter noise varies over time. (a) Polarization image corresponding to 𝑃 𝑆𝐴1 . (b) Polarization image corresponding to 𝑃 𝑆𝐴2 . (c) PDI component.
Fig. 12(b) that, the contrasts for PDI decrease from 0.6586 to 0.2365 slightly with increasing the optical depth of target. Comparing Fig. 12(a) with (b), a better image contrast can be obtained for PDI than that for direct imaging due to eliminating the scattering effect on the image. When the optical depth of target is 5.38, it is hard to discriminate the aluminum sheet from the wooden board for direct imaging. However, this discrimination could still be observed for PDI. 6. Summary In order to improve the efficiency of PDI in a scattering medium, we investigate the principle of PDI technique by means of Marius’s law and Mueller–Stokes formalism in this paper. PDI works based on the difference in polarization direction between the scatter noise and the target signal. When components of scatter noise at the orthogonal polarization axes of polarization state analyzer equal to each other, scattering effect is eliminated by use of common-mode rejection. It should be noted that if the scattering photons can be reused in the imaging configuration, the image quality can be enhanced further. Guo et al. ever proposed polarization retrieve method to reduce the scattering effect on the depolarization of polarized light through turbid media [31–33]. In the future work, we will introduce the polarization retrieve method into the PDI system to obtain better imaging performance.
Fig. 9. Experimental setup for PDI in turbid conditions. PSG: polarization state generator, MDP: micro-displacement platform, PSA: polarization state analyzer, A: aperture, L: imaging lens.
Eqs. (19) and (20) can be further sorted out as follows [ ] 𝐼PD (𝑖) = (𝑡2𝑥 − 𝑡2𝑦 ) cos 2𝛼 ⋅ 𝑠1 (𝑖) + sin 2𝛼 ⋅ 𝑠2 (𝑖) √ = (𝑡2𝑥 − 𝑡2𝑦 ) 𝑠21 (𝑖) + 𝑠22 (𝑖) √ √ √ ⎛√ ⎞ √ √ 𝑠21 (𝑖) 𝑠22 (𝑖) √ ⎜ × ⋅ cos 2𝛼 + √ ⋅ sin 2𝛼 ⎟ ⎜ 𝑠2 (𝑖) + 𝑠2 (𝑖) ⎟ 𝑠21 (𝑖) + 𝑠22 (𝑖) 1 2 ⎝ ⎠
(A.1)
where 𝑖 represents the component of scatter noise or target signal. On the basis of Eq. (3), Eq. (A.1) is changed into another formula √ [ ] 𝐼PD (𝑖) = (𝑡2𝑥 − 𝑡2𝑦 ) 𝑠21 (𝑖) + 𝑠22 (𝑖) cos 2𝛼𝑖 ⋅ cos 2𝛼 + sin 2𝛼𝑖 ⋅ sin 2𝛼 √ (A.2) = (𝑡2𝑥 − 𝑡2𝑦 ) 𝑠21 (𝑖) + 𝑠22 (𝑖) cos 2(𝛼 − 𝛼𝑖 )
Appendix We now show that PDI components of scatter noise and target signal obtained by Marius’s law and Mueller–Stokes formalism are identical with each other. 36
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Fig. 10. The distributions of angle of polarization for scatter noise and target signal at (a) 𝑂𝐷 = 2.37, (b) 𝑂𝐷 = 2.81, (c) 𝑂𝐷 = 3.46, (d) 𝑂𝐷 = 4.11, (e) 𝑂𝐷 = 4.69, (f) 𝑂𝐷 = 5.38.
Fig. 11. Image quality of PDI depends on the choice of polarization axis. (a) 𝜃 = 31◦ . (b) 𝜃 = 43◦ . (c) 𝜃 = 49◦ . (d) 𝜃 = 53◦ . (e) 𝜃 = 55◦ . (f) 𝜃 = 61◦ . (g) Numerical plots.
Fig. 12. Comparison of image contrasts using (a) direct imaging and (b) PDI. (C: image contrast).
where 𝛼𝑖 is the vector direction of scatter noise or target signal. According to geometric relationships in Figs. 2 and 3, the expression 𝛼 − 𝛼𝑖 is the angle between the direction of scatter noise or target signal and the polarization axis of polarization state analyzer. Comparing Eqs. (A.2), (9) and (10), the pair of Eqs. (9) and (19) or the pair of Eqs. (10) and (20) has identical expression. In this appendix, derivation is performed at the case of both linear elements of Stokes vector 𝑠1 and 𝑠2 being more than zero. It should be noted that Eq. (A.2) is still correct for other values of 𝑠1 and 𝑠2 , which is not repeated in this section.
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